I don't want to sound snippy, but you can do that easily with £1500 disposable a month.
I'm living on just under £1k a month before bills. And I have a car and a dog. I have an old house which is yet to see a month go by without having a tradesman have to visit.
I'm mortgage free, thank god, but still manage to pay the bills, go out for a meal with friends once or twice and save £250 a month towards things like Christmas, MOT (I move that money straight over so it's out of my bank account and into a separate one).
I'd recommend cancelling all unnecessary subscriptions. Switch your phone to a sim only contract if you can (I pay £8 a month for unlimited calls, texts and 16gb data). Check to see if you get a cheaper energy tariff at home and switch. Use cashback sites like Quidco when you shop online. I'd also suggest shopping online for groceries and get the cheapest possible delivery or click and collect slot. I go to a cashback site, choose my supermarket, click the link and the cashback usually covers the delivery fee - and seeing your online basket cost makes you review and ask if you really need the this and that which are £2 each.
Cut back on your Christmas gift purchases. Everyone will understand. For next Christmas, start saving in January. Remember the saying - people who mind don't matter and those who matter don't mind.
Obviously, I don't know how fast or urgent you need to get the £5k, but you can easily save £1k a month, at least, imo.