Chestnut I have been looking for a picture, but cannot find it. But our tree was roughly three times the size of yours. It was an ENORMOUS hazel. Oor tree surgeon had never seen such a big one. It had six intertwined trunks and was double the width of yours all round. It spanned about 30 feet wide. It had to be removed as all the trunks had started rotting. I am regrowing it from the boll.
We had a yew tree much the size of yours, or only slightly bigger removed about 10 years ago, that was well under £1,000. However in each case we have not had the roots grubbed out - and that does cost money.
Why not get a couple of estimates, then you will know how much it will cost, even if you do not go ahead with the job.