Can I live off £40,000 a year?

I'm with the others, it'll be incredibly difficult if not impossible in London.

Farringdon is in North London.  I'm in North East London and go to that area regularly which takes me about 25-30 minutes from my front door to Farringdon station.  Realistically I'd ride my bike though and get there in 20 minutes.  The surrounding areas are all incredibly expensive.  I'd love to live in Angel or Camden, just North of Farringdon!  Trace the train lines out and start to search in those areas.  The areas I can think of as known a a bit less but still nice are all realistically £1,500 / month minimum for a 3 bedroom flat, I just looked on rightmove and zoopla.  A friend commuted an hour each way Blackheath to Farringdon until Jan this year.  But I checked rent prices there and really, £2,000 / month and up for rent (utilities additional) for a 3 bedroom is the going rate.

For reference, I currently live in Zone 2, quite central, we have a 2 bed/2 bath with a balcony (no garden) in a modern block and pay about £2,300 / month in mortgage.  Rent was around £1,800 / month when we rented 4 years ago in the same area.  I used to live in Essex (east) and we paid £850(?) for a 2 bed/2 bath (small patio with shared garden, we were ground level) in that area 9-10 years ago but I had a 90+ minute commute each way and couldn't take it any longer.  It really was a get up at 6am, get home at 7:30-8pm, have dinner, go to bed and start over.  I was very cranky, tired, worn-out and bitter and it hurt my relationship with my husband.

With a £40,000 salary you'd need to compromise so severely that I think it'll effect your time with your family, your ability to settle in, your mental health or your bank balance that it wouldn't be worth it.  Unless your company can increase your salary substantially I'd be telling them to take a hike and start looking for a job elsewhere.

Working in Farringdon the OP has plenty of options to get into the city,  arriving into the likes of Kings Cross, Moorgate or Liverpool Street. If coming into Moorgate then you could get a flat in the likes of Palmers Green or Winchmore Hill. However you'd be looking to pay around £1400 for a 2 bedroom flat.

I would say you need to take schools into account as well if you have kids. I currently live in Stratford,  East London, and you can get a fairly decent 2 or even 3 bedroom flat or small house to rent around here for £1500 or less, and there are some decent schools here. It's zone 2 and the commute from West Ham station to Farringdon isn't bad. But it's not the loveliest  part of London to live in, it's very densely populated and there is a lot of poverty.  There are a lot of things to consider.