Alright, here's the actual design for an intergalactic mission to the Virgo Supercluster.
Phase 1: Acceleration
To begin with, you use really big lightsails and exawatt dyson-swarm-powered laser arrays to get your fleet of 30 or so ships (really just a cylinder of some fancy graphite-based dust-impact-resistant material that weighs about 1/5 of the Titanic, and has about a 20 meter radius) up to cruising speed of 0.9 c for their 200-million-light-year voyage across the intergalactic void to the Virgo Supercluster, or at least where it's projected to be in the future by cosmic evolution simulations.
Phase 2: Coasting
By time dilation, this is dropped to 100 million years of waiting in an absolutely black void between the galaxies, where nothing of note happens except for occasional nanobot repairs, and keeping the antimatter at 0.1 K. And most of the fleet dies because they got hit by a grain of sand that's out in the galactic void for some improbable reason, but over those sorts of distances, even very improbable sand grains will show up at some point. However, several of them probably make it through, with the front looking pretty moth-eaten.
Phase 3: Target Selection and the Steering Burn
At a few tens or hundreds of thousands of lightyears out, the next phase can begin. Telescopic monitoring of the incoming galaxy, to build up a map of where the stars will be upon arrival, and the interstellar density distribution, and pick a good-looking one. Sticking a telescope out in front leads to the sensors getting destroyed by the proton flux, so they'll probably be shielded at the bottom of a tube of solid-but-transparent material.
Steering to the appropriate star location is done by a dusty-plasma-fission rocket firing sideways, which provides 200 newtons of thrust (equivalent to a model rocket engine), and emits 3.5 gigawatts of waste heat. For thrust that low with that much energy, the exhaust must be going really fast, and by the rocket equation, it gets the 0.1% of c change in velocity with only about 5% of the starship mass devoted to propellant, ie fissile uranium (or plutonium, bred from ordinary uranium by an onboard nuclear reactor, which is much more common and less prone to decaying over these time intervals and easier to store). So that's another 7,200 tons of uranium. An antimatter beam core rocket would have much less radiation damage, but it only has 1/10th the thrust for the same power output, which may end up being a bit much for the radiators if we crank up the power by 10x to compensate.
To dissipate the heat, the cylinder extends diamond fins, which start glowing bright orange at temperatures that'd melt iron.
Phase 4: Magnetic Parachute Deceleration
At about 1.6 lightyears to go, near a peripheral star with gas density about 100x lower than the sun (and about 5000x higher than intergalactic space), the bulk of the dust shield is cut free to fly through the galaxy, leaving two sub-ships with a dust shield of about 14,700 tons each, or about 10 meters radius, which move away from each other and unfurl a 1 km radius (or larger if you want to do this braking maneuver in a more flexible range of gas densities, or brake over a longer distance) loop of superconducting coil behind them, charged up from the earlier dusty plasma rocket firing, dragging behind the dust shield, and attached by carbon nanotube cables (or some other really high tensile strength material, their mass is low enough that I think carbon nanotubes might turn out to be overkill)
These reduced dust shields will disintegrate, cutting more and more fragments loose, over the 6-year braking time, as lower speeds require a smaller dust shield, which also reduces the intercepted volume of space, and a smaller dust shield can decelerate faster. Think of the star as the enemy gate (it's down), with a parachute above you, and cutting mass off the bottom of the payload to fall below you so you're lighter.
With the superconducting loop parachute, and the rapidly shrinking dust shield, a peak deceleration of 1.5 g's is reached, which the antimatter storage is going to have to resist to prevent a big boom. Enough mass is lost on the deceleration to 2% of lightspeed that the dominant mass is from the magsail parachute itself.
Phase 5: Electric Sail Deceleration
At some point around 2% of lightspeed, the superconducting loop itself is cut free, and the next phase begins, with three more ships cut free and separating. Each consists of a dust shield a measly 2.3 feet in radius, with the payload, antimatter storage, antimatter reaction chamber, power-generating machinery, 1/4 of a kilometer liquid-metal-droplet radiator, and the machinery for the next deceleration strategy all hiding in that narrow cylinder of safe space, about 170x longer than it is wide.
For the next 50 years, the antimatter reactor works through its stash of about 160 g of antimatter, cycling liquid metal past the osmium ball and running a very compact turbine off of the temperature differential induced by the liquid metal, and cooling off the metal in the 10 megawatt radiator which makes up most of the length, so the spaceship would look like an arrow glowing red. This is to power the electric sail.
The electric sail consists of about 10,000 50 km long fine fibers, which are charged up to 4 million volts so they all stick out away from each other. Think of the spacecraft as a dandelion seed with a really disproportionate parachute, 200x longer than the seed tail length. This electric sail repels protons, which causes deceleration, but they work way better at low velocities than magsails. Steering can finally be done by charging different fibers by different amounts. The long narrow radiator is in tension, not compression, since the electric sail is at the tail, so the ship can safely be that lone without collapsing. The charged fibers attract electrons, so we'll need a 1.5 megawatt electron gun at the tail. Total spacecraft weight at this stage is 18 tons, most of which is the radiator, antimatter reactor, and dust shield.
Over 50 years, this decelerates to 600 m/s near some suitable asteroid or comet.
Phase 6: Final Landing
Finally, the last 0.9 ton stage detaches to do a final landing.
600 m/s of change in velocity is easily attainable by conventional chemical rockets. We aren't taking off from orbit, we're dropping onto an asteroid, so the rocket of choice is the monomethyhydrazine-dinitrogen tetroxide thrusters that are used to alter the orientation of the space shuttle in orbit. Those rocket engines only weigh a couple kg, and the specific impulse is high enough that we only need 18% of the rocket mass composed of propellant, or 160 kg. The final stage lands on the asteroid or comet, and deploys the Von Neumann probes, the frozen state of the emulated people who decided to come along (they can't be active for the voyage because most of the ships are going to die by dust), and either some solar panels for probe energy, or a 490 kg mini-nuclear reactor for probe recharging if it's far from the sun (that ice isn't gonna melt itself...)
And done! No 700 kilometer antimatter starships needed.
Just a multi-exawatt dyson swarm laser array, some ridiculously big lightsails, giant blocks of graphite launched at 0.9 c, multi-million-year antimatter storage, a few thousand tons of fissile material to fire a nuclear rocket for thousands of years as steering, diamond radiators, superconducting magnetic parachutes, several tons of osmium, a few kilograms of antimatter, a quarter-km of liquid metal radiators, a 50-km radius "cosmic dandelion seed" of thin fibers charged to several million volts, good-old-fashioned chemical rockets, self-replicating probes, and repeated spacecraft sacrifices to the vengeful god of cosmic dust, may future civilizations punch him in the metaphorical face by making a design better than this, I know it's doable
Finally I can exorcise this special interest from my soul and get back to the fancier sort of math that ensures I live to see it happen.
(Also, Anders Sandberg, if you're reading this, hit me up, I've got an unrelated paper idea you might be interested in)