Learn Polish: Nasz lekarz mieszka niedaleko szpitala. - Our doctor lives near the hospital.

The preposition niedaleko (near, not far from) requires the noun that follows it to take the Genitive case. This is a strict rule for several prepositions of location in Polish.

I know that szpital is a masculine inanimate noun. Why does its Genitive form end in -a instead of -u?

That is a great catch! While most masculine inanimate nouns take the -u ending in the Genitive case, there are many exceptions that take -a instead. Szpital (hospital) is one of these common exceptions, so it becomes szpitala.

Does niedaleko literally mean "not far"?

Exactly! It is made up of nie (not) and daleko (far). When used as a preposition meaning "near" or "close to", it is always written together as the single word niedaleko.

Why do we use nasz here instead of naszego or another form?

Nasz (our) describes the subject of the sentence, lekarz (doctor). Because the doctor is the one doing the living, the noun is in the basic Nominative case. Since lekarz is a masculine noun, its matching pronoun is the basic masculine form nasz.