Adopt ESLint rules to prevent unsafe use of `any`

[felixfbecker]

These are recent additions to @typescript-eslint. Will potentially require refactoring, but will make our codebase safer.

Some major cases where we currently use any:

• catch clauses. We should use type guards to check the error type and asError() if we want the message.
• Custom type guards. This is currently difficult because the in operator cannot narrow types from unknown. Composing type guards would work though, we should look into adopting a library like https://github.com/sindresorhus/is (and make sure it has functions and types expressive enough for us)
• HTTP responses. We should use explicit casts.