If you wanted to run a 5 year old Dallara in Club F3, for example, near the back of the grid (i.e. saving money where possible, and doing as much on your own as you can), then your annual budget will be around £30,000 - £60,000, not including the cost of the car, or the hire of extras.
This is a series (not a championship), and you won't 'get any further' by being spotted in it.
If you want to race something like Monoposto or F4, then you could (using the present rules) get a car (between £5k and £30k), and do a season of racing on about £10k.
Parts are expensive (Dallaras) or virtually impossible to buy (Reynard, Ralt etc) unless you find someone with some second hand parts. Therefore you'll need to be able to make stuff yourself, or find a company that is happy to make one offs as and when to short deadlines - not cheap.
Preparing the car yourself - there is NOTHING special about single seaters. They obey the same laws of physics as a road car. If you know about cars there is no reason why some threads on a racing car will be any different - though of course their sensitivity to change might be in a different league!
This is for me:
The car was bought entirely for the hell of it, with no intentions to use it in competition. It was between £5k and £10k as seen (running).
The 'team' consists of my father and myself. We do everything from engine changes/overhauls to making brackets to wiring it, to running it on the day, to setting it up and, of course, driving it. We had no prior knowledge of single seaters, but loads of engineering and road cars.
Our budget isn't really known - it's simply the least we can get away with without giving up huge advantages - we run the hardest compound on the grid to save buying new tyres for example. I would estimate that the running of the car costs between £2k and £5k per year.
Entry fees are around £200 per race, and there are 13 rounds - so £2600 for that.
Travel costs are probably about the same as the entry fees etc.
I'd put our annual budget at an unofficial £10k to £15k all in, but if we can save money within reason we will. Having said that, pit monitors, pit boards, onboard cameras, laptiming displays etc are non-essential addons, but ones we felt gave a large assistance
rice ratio, or were particularly interesting/educational.
Only part of our remit is the racing - the fact that we put in over 150 hours in the garage prior to EACH race demonstrates that the engineering side of things is very important - if we didn't enjoy preparing the car we wouldn't bother going racing.
My social life is in tatters (though I try to squeeze in one evening a fortnight or three weeks for friends. My bank balance is pretty much dead - every spare penny I earn goes into the racing. Additional funds that I cannot afford are then requested (but not guaranteed) from my father. I work 9 - 6 at work, then 6:30 - 10:00 on the race car pretty much every night. This is for a 15 minute practice and a 15 minute race with no chance of climbing any ladders within the sport, or any prospect of being paid to do it.
And it's ABSOLUTELY FECKING WORTH IT!