Is a 6000w amp safe to run at home?

If the device has a regular mains plug on it then it will not draw more current than an outlet can safely deliver. It is possible that by connecting the wrong speakers to it, you could fry the power section of the amp, but that will not cause a fire, it just means some transistors could overheat and then the amp will stop working. Overloading an amp could also kill the speakers, but they wouldn't catch fire. There will probably be a nasty smell if the amp or speakers are damaged.

The amp will work just fine. As Andre said the true power rating is way less than that. If it is a iNuke NU6000 the true power rating is 300 watts into four ohms per channel.

Behringer actually makes some pretty good stuff in many cases. I have a pair of Behringer amps at my church that have been running 24/7 since 2008.

Safe? Probably. Its a cheap class D amp, Who's maker does not post specifications for it properly, which always means the specs are bad to horrible. The power ratings posted are questionable though not impossible for a class D instrument/stage amp, if all specifications for distortion, or accuracy are ignored, as they are with this product. Likely a poor performer for home speaker applications where accuracy and lack of background hiss are considered essential.

One of the things wrong with such bargain basement companies is that they grossly LIE about the actual audio power of their products. Rest assured that this amp in NO way puts out 6,000 watts.

A PROPER power amp gives it's power in the following manner: X watts PER CHANNEL, BOTH/ALL channels driven, 20-20K Hz, 8 ohms, with a total harmonic distortion of no more than 0.1%.

That would depend on the input power requirements of the amp and what your house power is rated at....and since I see none of that info in your question.....It's hard tellin', not knowin'.....

My brother has a 5000w amp and it runs fine.