I'm amused at the idea that anything 3 miles from Winchester could be described as remote!
However, more seriously, do you have any experience of living in the countryside or are you among the south-west London exodus to Winchester and surrounds? DH and I did a similar move a few years ago from suburbia to a similar-ish (but smaller!) house closer to the coast. DH loves it because although he grew up mostly in cities he is very sporty and outdoorsy and this gives him the time and opportunity to pursue those activities. I am probably more in the camp of tolerating it because I have "muscle memory" of growing up in a similar environment.
The big drawbacks for me are broadband speed/mobile reception. We have no mobile signal here and when we were both working could only manage two conference calls and nothing else - anyone else in the house needed not to stream or download or even use their phone! I can't imagine you will have the same difficulties but do check because it impacts every part of our lives nowadays:
no Deliveroo, Uber etc. Doesn't bother me too much but it may be a shock if it's been part of your life;
Having to get in the car for EVERYTHING. Like you I could walk previously to our local train station, doctor, newsagent, various friends, coffee shops etc. Everything takes a lot more organising now. I always have to drive myself to every book club or evening out because people live over a more scattered area and there's no public transport. Check the taxi situation as well. I'm sure being that close to Winchester it shouldn't be a problem but useful to check that you don't need to book one (as you do here often) days in advance.
On the house itself, yes it will cost a fortune to heat and you may not be too happy about sitting on the ground floor in the winter evenings staring out at a wall of blackness. I would hate it but I think it would be tricky and mega-expensive to arrange any curtains/blinds for that expanse. Also, coming from suburbia to what looks like working fields around, be prepared for a lot more dust and insects.
No idea on the price as I don't know that area well enough but Winchester in general is pricy and attracts people with deep pockets.