Storing lemon juice with crushed mint

I'd make it into ice cubes.
They'd add decorative interest too.

Fridge, maybe a week, freezer, more like 6 months.

There's a full list of storage times in How long can I store a food in the pantry, refrigerator, or freezer?

I would like to add that you can just freeze the mint leaves without grinding them too. Wash and dry them well, then... Put them in the freezer. If they're fresh and dry, then they won't stick together, and you can pull out as many as you like at a time.

That said, if you let the leaves thaw, they'll look and feel like they've been blanched. The water in the leaves will burst the cell walls while freezing, as surely as it would while boiling.

I've had experiences in the past where herbs + lemon juice ended up getting a kind of "pickled" taste because of the acidity of the lemon. I might just grind the mint and freeze that on its own, and freeze lemon on its own, without combining the two.

Another idea: Make a tincture/ cold infusion.

Wash and dry your lemons and mint. Dice and freeze the lemons; freeze the sprigs whole in a bag. Beat the bag about a bit to detach the leaves. Place frozen lemon chunks and frozen leaves in blender. Whiz to powder. To make a cold infusion, add water; to make a tincture, add vodka (well, not really a tincture, due to the high juice content). Whiz some more. Strain out the solids by pushing the goop from the blender through a sieve with the back of a ladle; use a muslin cloth or simply your bare hands to squeeze out the last drops from the pulp. Bottle the liquid and keep it in the fridge. Fine particles will settle out over time, yielding a clearer liquid. Add to drinks as required.

The quasi-tincture will have a long fridge-life due to its alcohol content. The infusion not so, but this can be helped by adding sugar or super-concentrated syrup at the time of bottling.

Thank you to rumtscho for helping me to improve this answer in terms of clarifying its relevance to the question, partly by expanding upon the other respondents' reasons for having suggesting alternative strategies (freezing rather than refrigerating; not combining reactive ingredients).

The answer to your question as posed (storing the two ingredients, mixed, in the FRIDGE) is about 3 or 4 days, or as Tetsujin advises, max. a week. If you're doing this often, that may be sufficient for your needs. It is worth noting, however, that the period of freshness would depend on a) the temperature setting of your fridge, b) how closely sealed the container is, and c) how much air is also inside said container (the more there is, the faster the contents will oxidise). Best strategy would therefore be to use the smallest possible container for the amount you have to store, seal it well, and keep your fridge relatively cold. Taste to test.

That said, all four answers offered so far (including this one), ultimately recommend not using a fridge but a FREEZER instead. kitukwfyer also recommends leaving your mint leaves whole until you're ready to use them, rather than crushing them, to whose prescription of making sure the leaves are fully dry (as in 'not wet', not as in 'dried herbs'-dry) when you freeze them, I'd further specify freezing them in a bag rather than in a rigid container. You can then crush them, still frozen, by whacking the bag a few times on the countertop; the leaves will detach from the stems (saving much time-consuming leaf-picking, assuming your desire for 'crushed' mint precludes the stems) and naturally crumble into a coarse powder or meal which you can then shake out easily into a glass or their other ultimate destination without them melting and turning into mush.

The ice-cubes-alternative has the advantage of making your drink sharper-tasting as the juice-cubes melt, which can be balanced by not-quite-fully-mixing any syrups you may also be adding, so the whole thing becomes more concentrated as you imbibe - the opposite of the watery dregs that sipping too slowly can leave one with when using pure ice.

I'd also second Esther's answer, which suggests storing the two ingredients separately. Different foods tend to interact, and you might end up with something more akin to salsa than cocktail aromatics.

So if 'three or four days, up to a week' doesn't meet your hopes, if you possess a freezer, and you don't mind performing the very last steps of processing your carefully-preserved ingredients at drinks-mixing time, here's what I'd do...

Lemons: Freeze them whole, fresh from the shops. Take a lemon out of the freezer and let it soften slightly (about four or five minutes). Use a very sturdy knife (not your best one!), pare off a flat bit on one side so that it sits stably on the board, and then pare off thin slices from one end. Unused lemon goes back in freezer for next time. Personally, I use the flesh, juice, peel, pith, and seeds together; you'll get more flavour (if not more acidity) from each lemon, this way. For a drink without solid particles floating about in it (besides the mint), you can use a pestle or other blunt object to muddle the parings, now fully defrosted, through a strainer to extract the juice, suffused with zesty fragrance and ultra-fresh.

Mint: Store it alive, on the plant. I know this goes a long way outside the ambit of your original question, but the more flavourful result makes it a solution worth considering. That said, the freeze-and-shake-in-a-bag method isn't bad at all, and great for dealing with large amounts at one time.