Can you keep a lobster for a pet?

Well you could, but you are looking at setting up a BIG coldwater marine tank, filters and probably a chiller to keep the water cool.

More practical.

Get a freshwater crayfish. They are smaller, and can live in normal freshwater, cold or tropical depending on the species.

They need a normal filtered fish tank, but a 10gal tank with a freshwater cray is perfectly practical, compared to a 200gal marine tank anyway.