Any 12v battery would work. Lead acid deep cycle batteries probably represent the best value and best compromise in terms of life expectancy, capacity and price. Which one you pick will determine how long you run your equipment.
To get a very rough idea of how much wattage a battery will supply, multiply the voltage by the capacity (amperage).
This battery has a capacity of 33Ah (amp hours). In this case, you get right around 400Wh (33Ah x 12V = 396 Wh). Again, this is a rough estimate, but it's close enough for the purpose of our discussion. I've looked around and it appears a Gyre will use around 25W, depending on the model and speed. You haven't mentioned your return pump's model, but let's assume that's another 25W. So together these two pumps will use about 50 watts per hour. So, our battery of 400W will run these two for about 8 hours (400W / 50W = 8 hours). If you turn down the Gyre (and the return pump, if it's DC), you could get even more time since the power usage is reduced. To get more capacity, you could order two (or more) of these batteries and wire them together, or you could buy a larger battery.
This 100Ah model is more expensive (around $160), but it offers up to 24 hours of capacity (100Ah x 12v = 1,200Wh, 1,200W / 50W = 24 hours). You could also only run the Gyre and leave the return pump out of the equation, which would roughly double these run times.
One thing to keep in mind, these are theoretical maximums. The second battery is rated at 100Ah (1,200 Wh), but that's if you discharge it all the way to zero, which I don't think is great for your batteries long term. Additionally, the act of transforming power from DC to AC (then back to DC again, for pumps like the Gyre) will also lose energy to various inefficiencies. So if your actual power use is 50W between your return pump and your Gyre, the 100Ah batter will likely get you less than 24 hours of run-time.
You've probably noticed that this approach is not necessarily cheap. The inverter mentioned above and the 100Ah battery would cost you more than $300. At that point, it may be worth looking into a generator depending on your situation. I, however, like the inverter route for several reasons. First, it's automatic. You don't need to lug out a generator, start it up, and plug all your aquarium equipment into it. On failure, the loads switch over to battery and your Gyre and/or return pump keep running as if nothing happened. Second, a generator, while it may be cheaper, requires maintenance and fuel. This battery backup system, depending on battery shelf life, could last 10 years or more with no maintenance at all. Third, in my area very few outages go for more than a few hours. Only twice in the last decade have I had outages more than 24 hours. Depending on your circumstances, a battery system like the one we're discussing may mean you would never need a generator. Again, it depends on how long the outages last in your area. At the very least, if you do have long outages, you have time (24 hours) to find a generator. Fourth, if you're like me and live in an apartment or on the second story, you may not be able to have a generator at all. A battery backup system may be your only choice.
If all your pumps are DC, it might be cheaper to look into an IceCap or Ecotech battery backup solution. You just won't be able to run any AC equipment off a dedicated IceCap or Ecotech battery. If you feel like doing a bit of DIY, you could make an
DC battery backup clone with a battery and some wires. You'll just need to include
a charger, but they're pretty cheap.
