Learn Spanish (Spain): Necesito medio litro de aceite.

In Spanish, you don't use the article un or una before medio when talking about measurements. While in English you say "half a liter", in Spanish it is simply medio litro (literally "half liter").

Can I say la mitad de un litro instead of medio litro?

You could, but it sounds much less natural. Mitad is a noun meaning "the half" (used for things like la mitad de la pizza). For standard measurements like liters, kilos, or hours, native speakers almost always use the adjective medio (half).

Does medio change if the noun is feminine?

Yes! Here it is medio because litro is a masculine noun. If you were asking for half a bottle (botella, which is feminine), you would change the ending to match: media botella.

In Spain, does aceite automatically mean olive oil?

Not automatically, as aceite is the general word for any kind of oil. However, olive oil (aceite de oliva) is the standard cooking oil in Spain, so if you ask for aceite in a food context, that is almost certainly what you will get!