Learn Russian: Учёба занимает всё моё время.

Although время ends in -я (which usually signals a feminine noun), it is one of the special Russian nouns ending in -мя (just like имя, meaning "name"). All -мя nouns are neuter, so "all" and "my" must take their neuter forms: всё and моё.

Is всё моё время in the accusative case here?

Yes, it is the direct object of the verb занимает (takes up). It doesn't look like it changed because inanimate neuter nouns (and their modifiers like всё and моё) look exactly the same in the accusative case as they do in the nominative (dictionary) case.

Can the verb занимать be used for things other than time?

Definitely. Занимать (imperfective) and its perfective pair занять mean "to occupy" or "to take up". You can use it for taking up physical space, occupying a seat, or even borrowing money.

Why use the noun учёба instead of a verb like учиться?