Learn Turkish: Oda moda temizlemeyeli hiç ferahlamadım. - I haven't felt relieved at all since I last cleaned rooms and stuff.

This is vague categorical reduplication. By taking the noun oda (room) and repeating it with an m at the start to make moda, it creates the meaning "rooms and stuff" or "rooms and such". It implies a general category of chores rather than just cleaning one specific room.

Why is the verb temizlemeyeli negative if the translation says "since I last cleaned"?

The suffix -(y)eli means "since [doing something]". When you add it to the negative form temizleme-, it literally means "since I have not cleaned". In English, we naturally phrase this as "since I last cleaned", but Turkish logically measures the duration of time that you have not been doing the action.

Why isn't there an accusative case ending on oda moda? Shouldn't it be odayı modayı?

You would only use the accusative case (odayı modayı) if you were talking about specific, definite rooms (e.g., "since I last cleaned the rooms and stuff"). Because the speaker is talking about general, non-specific "room-cleaning" as an activity, the object stays in its bare dictionary form.

What kind of "relief" does ferahlamak refer to here?

ferahlamak comes from the adjective ferah (spacious, open, airy). While it can describe a physical space becoming less cramped, it is very commonly used for a mental "breath of fresh air" or the psychological relief you feel when a burden is lifted—like the satisfying, relaxing feeling of finally sitting down in a freshly cleaned house.