The title pretty much says it all.
I'll be leaving from near the National History Museum, so the closest stations are Gloucester Road and South Kensington.
My objective is to minimize cost.
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The simple answer to your question is you cannot use a Z1-4 travelcard to do the journey you request. Gatwick is not actually in London. But you are actually asking how far you can get on the Z1-4 travelcard
The simplest and almost cheapest option is going to be to use the Thameslink service from Farringdon or Blackfriars to Gatwick. Your Z1-4 will get you to either of those two statoins on the circle line, and as at May 2018 a single to Gatwick from either station is just under £11. And it is a simple one station change.
A z1-4 will get you to east croydon, but only via London overground. And then you have a one or two train journey onwards, with an additional fare of just over £5
Victoria to Gatwick as of May 2018 is around £17, on regular trains, with the Gatwick express attracting a premium fare of just over £20.
In looking at these prices I have assumed that you will be traveling during off peak times.
There is a balance of convenience verses price verses length of journey. And Thameslink seems to be a happy medium for what you are requesting
I haven't noticed anyone give this answer yet, so I'll give it myself:
You can buy a Boundary Zone 4 to Gatwick Airport fare for £6.60 (single with no railcard). You won't be able to buy this online, and most ticket machines won't sell it either, so you'll probably need to ask at a ticket office. This will be valid only in conjunction with a Zone 1-4 Travelcard on your Oyster.
You should be able to use this on any service from London to Gatwick, even one that doesn't stop in Zone 4; just use your Oyster at the barriers at the London end (if you have a travelcard you won't be charged) and your Boundary Zone ticket at the Gatwick end. Probably using the pedestrian subway to South Kensington and then going on the tube to Victoria and catching the Gatwick Express will be easiest.
If you really want to save cost as much as possible you can go to East Croydon as mentioned in other answers, but you can only do this via the Tram, as East Croydon is only in zone 4 for tram purposes (otherwise it's in zone 5). I would highly recommend against this as the district line to Wimbledon is supremely slow, as is the tram itself, by comparison to heavy rail options. But if you do you'd probably end up buying an East Croydon to Gatwick Airport ticket for £5.30 (any train to Gatwick) or £4.60 (Thameslink trains only). All that extra time to save £2? You'll have to make your own mind up; maybe it sounds good to you, but I wouldn't do that.
If you're looking for a route to the nearest point then East Croydon is where you'll end up, taking the District line from either of your starting stations, west to Wimbledon, then changing to a tram for the ride to East Croydon Station. You can get a train direct to Gatwick from there if that's your real objective.
However, it's a long and slow route on suburban commuter services. A faster and simpler route would be to head east from your starting point to Victoria mainline station and catch the Gatwick Express train direct to the airport.
You could also take a regular mainline train from Victoria to Gatwick which will take a bit longer but might be cheaper.
There's another more involved route via Whitechapel and West Croydon if you're in the mood for adventure. I'm sure you can find others. See the London Zones Map available from the Transport for London web site.
Under a fairly specific set of conditions, (you're willing to book in advance and commit to travelling at a specific time, while still being willing to be slightly flexible about what that time actually is) then it's possible that one of the coach services from Victoria coach station will end up cheapest. There are certainly services advertised by "easyBus", but I believe run by National Express advertising £2 single tickets from London to Gatwick. However, from what I've heard, journey times are longer than any of the train routes, and the service can be crowded.