Is it possible to ask about RA positions after being rejected by the same lab for a PhD?
You can ask for employment, but the title and work would probably be different from that of an RA. And the opportunity for authorship of papers might not be available.
In some fields, labs are very large, usually grant funded. There may be more need for lab assistants than can be fulfilled by graduate students. Someone has to watch that the kettle doesn't boil and bubble too much over night.
Holding such a position could possibly lead to interactions that open additional possibilities. But you would be an ordinary employee, subject to rules and protections for employees.
For an extreme example, I doubt that everyone working on projects at CERN is a student or professor. On the other hand, some at least, of these people wind up on long long authorship lists when experiments succeed.