The C-style "oops I used an object after freeing it and now anyone can execute arbitrary code" style of vulnerabilities is confined to a fairly narrow set of programming languages, most notably C and C++, which unfortunately happen to be popular and to have some interoperability advantages. One of the key requirements of a security-amenable language is that it can never tempt its users into writing parts of their project in C, which happens if the language is too slow (eg Python) or can't otherwise interoperate with important systems (most languages, unfortunately). Some programming language choices create performance and sysadmin problems that are treated by end-users as dealbreakers; in fact this was responsible for most of C's historical success.
A lot of issues crop up at the interfaces between programming languages, which tend not to fall neatly into one of the languages' scope. SQL injection is a classic example. A language can't do a whole lot in practice to protect against that, but a library can: by designing the APIs so that doing things the safe way is easy, and doing things the dangerous way requires calling functions with "dangerous" in the name (like React's dangrouslySetInnerHTML).
If I import third-party libraries from the internet, there is a lot of opportunity for mischief (both by those third parties, and by the people who might hack them). This is probably the nodejs ecosystem's weakest link at the moment. This problem is largely social; the best defense is a trustworthy group of curators providing a core set of libraries that you rarely need to step outside, plus an expectation that libraries are large projects that avoid creating indirect dependencies and that you use a small number of. (As opposed to having a hundred different left-pad style tiny libraries from a hundred different authors.)
Reducing security vulnerability incidence to zero is possible in some domains, but a programming language alone can't do it; I can write if(password=="backdoor") acceptLogin() in any language.
I think there's a fair amount of room left for incremental improvement, but in practice I think it looks less like "move everyone to Haskell and Coq" and more like "design a good core-library crypto API for X" and "reform common practices around npm".
It is already possible to build an embedded language inside of Coq which requires arbitrary security proofs for all executable code. It's theoretically possible to make those proofs guard against all known side channels, including, theoretically, hardware-based side channels such as Row Hammer.
Are you asking about which kinds of attacks can't be stopped by improving software? Or are you asking about the theoretical limits of PL technology? The latter question is not so interesting without more constraints: as stated above, they're basically unbounded.
Given the most security-amenable language you know, or that you could imagine building (or given some tools and processes that change the qualities of an existing language), by what proportion could we reduce the incidence rate of major security errors? In what domains could incidence be reduced all the way to zero?
Are there situations where your more secure language can't currently be used, because it imposes higher development or runtime costs?