EDIT: Due to numerous changes to the original post, this answer no longer really applies. Keeping for its comments' sakes.
You'll probably want to start with a 12V battery, and use a 12V to 5V "buck" converter to supply your 5V. One that can handle 5A should be 10-15 USD, but you might want to go with a higher rating if this 4A load is constant, to give some room for error. Then figure out how long you want your equipment to run for - if you want a 10 hour backup, you need 4 * 5 * 10 = 200 Wh of capacity. For a 12V battery, if you only use the top 40% of it's capacity (for long life), you'll need a 40 amp-hour battery, ([200 / 12] * 2.5 = 41) which should run you maybe 60-70 USD. Also, when you wire all this up, make sure your connection to your equipment is the same polarity as the charger - the symbol on it says it's center-positive.
12V batteries come in a few types - AGM, SLA, Marine, car battery... I've had good luck with SLA, or Sealed Lead-Acid, myself: they are well-contained, last for 5-7 years of daily use if properly sized, and charging is simple. If your equipment is mobile, you could try some kind of Lithium rechargeable battery, you'll get good energy density, but you'll have to deal with the risk of exploding batteries, complicated charging, and a shorter lifespan.
Don't use a car battery - it will likely die after only 10 or so full charging cycles.
Lead-acid batteries have two parts to their "capacity": total amp-hours, and discharge, or "C" rate. Multiply the C-rate by the amp-hour capacity of the battery to get the amp draw that that rating applies for. For, say, a rating of 40Ah @ 0.1C, the battery will supply 40 amp-hours if you draw 4 amps continuously. The general rule of thumb is that the more current you pull out of the battery, the less amp-hours it holds. batteryuniversity.com has some excellent in-depth explanations of all this.