We did, with a 4 month and nearly 3 year old. It was a year of hell... boiling hot in the summer and freezing in winter!
We bought a 3 bed 34f x 10f for £2000 from a holiday park. We could of picked up one much cheaper but they needed new carpets and just looked tatty. When you look at how much you save in rent it is probably worth it though! We lived in ours while we were doing a new build. Be prepared to fall out with your neighbours..
We also bought ex-site (but it was a privately owned caravan so in pretty excellent nic). We also had a small budget around £3.5k. We fitted carpet but made the mistake of a beige and should of gone for a cheapo darker one. At the time there was 2 of us and a newborn so only 2 beds. We had to go for a small one as we had to get it through the lanes (must check that first!) It was the narrowest and 28ft long - Consalt? I bought cheap curtains from Dunlem chopped them down and lined them.
Shouldnt think you can rent, not only the cost but they get wrecked! ours was discusting in the end, the dog, chickens and wellies were in and out, when it rained and you opened the door it came in. It stunk and its in the field now waiting to be burnt.
We had gas bottles (always have two), the toilet waste went into a tank that hubby found and always over flowed (we pumped it out with tractor and slurry tank borrowed off friend) The elec wasnt a problem.
Yes agree its not pleasant, it took me all day to wash up as there was never any space however, I cooked proper meals every day! We had a shed next to it with freezer and additional fridge plus shelving for tons of cooking stuff.
It was freezing in winter and we sat against the drafty windows which had mass condensation - we bought loads of those oil filled rads which we very good. In the end the front door came off its hinges and we had to use string to shut it (did have some laughs), The baby used to fall out reguarly.
Its worth it for a short while but you have to lower yr standards and be very organised.
Cost wise its the only alternative - you go away from site and start renting a property including all the services - you'd be wasting £1.6k per month while a caravan would be incrediably cheap. If your doing renovations to a high standard including a listed building you need to double or triple the costs of a standard build. You need every penny and from our experience the renovations are money and energy zapping.
Lots of people have lived well in a caravan and know loads who live full time.