Broadly speaking, any time an economy moves beyond bartering of goods (my wheat for some cloth etc...) and into a money system, that system can only work because of a mutual trust and shared agreement on the value of that money.
We are so used to this mindset it is hard to escape; we think of gold, gems and other 'obvious' currencies for a fictional world as having inherent value as tokens of worth.
So lets move away from gold and say that the city uses banknotes. The Gods have an excess of any mere physical good, so gold is a useless token of trade (anyone can gather ridiculous amounts of it with ease). Instead they agreed upon a value tied to banknotes, such objects are thus valuable in their economy, but useless anywhere else in the world because people don't recognise their purchasing power
2. Wireless power
If you absolutely need a physical object which degrades outside the city via science, then best bet are small money tokens that appear to be glowing rocks. The glow is in fact the result of microelectronics where an internal wireless receiver in the rock gets energy from the precisely-tuned EM field that permeates the city, and that receiver powers a light source (different colours for different 'coin' values?).
Let us assume that all the circuitry is incredibly fine, and essentially invisible to the human eye, even if the rock-tokens are split open.
As in the banknote answer, these glowing tokens are an agreed upon money system in the God's city, backed by a banking system. Unlike in the banknote answer, glowing rocks look amazing and clearly are worth stealing to be sold elsewhere for a lot of money.
However, removed from the EM field in the city, these tokens would stop glowing and appear to just be meaningless pebbles. Thus become worthless to the uninformed observer.
Further interesting lines of exploration would be Gods out in the world with battery-powered units that can make the coins glow, and thus the Gods can trade successfully (and in secret) with their currency outside of their main city.
An alternative that works on a similar principle could be fluorescent ink -- the city is permeated with harmless non-visible light of a particular wavelength, and you have rock/wood tokens covered in an ink that fluoresces in the visible spectrum in response to this light. The tokens will become superficially useless when removed from these lights (although this method is harder for someone to be ignorant of, we might expect tokens to go dark in pockets, boxes, dark rooms etc... prior to being removed from the city)
3. "Actual" Gold that Degrades
[added in edit] This is hard. I don't know if there is any actual method to do this. But first we'll define the problem: reflectance & weight.
A gold-esque substance needs to be heavy and characteristically shiny above 500 nm. Potential ways I can imagine doing this reversibly are:
- suspend tiny particles with a characteristic lengthscale & spacing in a resin such that you engineer the approximate reflectance, then have an environment that destroys the resin outside of the city, leaving behind non-golden dust and liquid (sonic resonance? a large magnetic field that yanks or inductively heats the particles?). Unfortunately, suspensions in resin are unlikely to have the bright sheen of a real metal's surface.
- Use a metamaterial of variable reflectance that is activated by electric fields (or an applied voltage difference from circuitry) in the same way as the glowing pebbles are in my second answer. Such materials are barely in early testing (Some Graphene research). I don't know how well such metamaterials could plausibly be engineered to look like gold, but it might pass a sci-fi sniff test.
As we can see, these "golds" are not easy to back with hard science, but they might pass muster in a story with sufficient handwaving. Options 1 & 2 are scientifically stronger and can meet story requirements unless gold is specifically needed.