Uhm IIRC typical battery is 2000 mAh/day, so one week battery is 14 Ah/day, which is 50 Wh assuming 3.7 V.

A typical sodium ion battery (which i very much like btw) typically holds 0.2 kWh/kg, so 200 Wh/kg, so to store 50 Wh, you'd need around 250g of battery.

For reference, i think smartphones should be about as heavy as an apple (fruit) which is 100g average. And the battery makes most of that weight (like, 80%). So the battery could be about 80g, which would store 16 Wh of energy. That would make about 4000 mAh. Which is what many phones today already have. Which lasts for 1-2 days.