Staying at Gatwick

This is in the Montenegro forum. I’d still advise against it in the U.K. If your budget is that tight, pick a hostel or else skip London. London is one of the most expensive cities in the world.

It could take up to two hours to travel from Gatwick to central London, particularly if the M25 is heavily congested (which it mostly is) plus factoring in the cost of daily travel and you'll find staying at Gatwick to be a false economy.

The price is exceptionally low for a reason - they don't expect much custom over that period for the reasons above. It would normally be around double that on a weekday

staying at Gawick Airpot . . . . take the bus to the city center.

Assuming by "city centre" you mean London, taking a bus from Gatwick (or anywhere else) into central London is not a good idea. Train is faster, more comfortable and more reliable (no traffic delays).

And staying at Gatwick and commuting into London is a bad idea for all the reasons the others say.

for 5 days during the holidays

Which holidays? When? In the UK, the phrase "Holiday period" usually refers to the summer, when everybody goes on holiday.
You don't say which country you are from. I understand that in some countries the phrase "the holidays" confusingly refers to Christmas.
If you mean this, be very aware that everything shuts down on the 25th December, no transport, no restaurants without advance booking. You have been warned.

Thank you all for your suggestions. I meant the Christmas. I’ll look for other accommodations, thanks for the input!

Just to clarify, if clarification is required, that when people say no transportation on Christmas Day, they mean none. Zero, Not "just a little". There are no trains on Christmas Day, and all the shops and restaurants except a very few restaurants and pubs run by non celebrants. They jack their prices up, and you need reservations. You may be a little late to get reservations now that we are only 2 weeks out.

There are tubes and buses on a very limited schedule on Boxing Day, but still no trains.

Are you coming from Montenegro?