More or less random pick from Amazon, $18.99 before coupon. It can charge from USB or solar (slow) and output up to 3A at 5V. It’s rated at 38,800 mAh.
Here’s the plot. Charge up this sucker and then hook it up to a weather alert radio. Let’s say the radio consumes 100 mA (at 5V), then this battery would power the radio for 388 hours or 16 days — more than 2 weeks. The radio should be a able to power itself while this battery is recharging.