Then you could use a UPS system. I have found this chap on ebay who sell reconditioned units with brand new batteries. http://stores.shop.ebay.co.uk/UPS-Trader__W0QQ_armrsZ1 .
I ordered a UPS700 from him for my master clock system as it needs 12v and 6v batt backup.It arrived next day.
All you do is plug your weather station and meteohub server into the mains plugs on the unit and on mains fail the unit converts the 24v standby battery back up to 240volts keeping your weather station running. The unit will auto shut down when the batteries are flat (unlike my design). There is also a plugin module £15 so you can check your UPS over the internet. There are more powerful units from the same manufacturer if required.
You could also plug your internet router into the UPS if you want to keep your server "live" on the net during blackouts.
The only drawback is my UPS uses around 21watts of power so with your server and weather station plugged in you are looking at approx 30 watts total. My DIY batt backup uses around 12 watts all in.

Full details on the APC UPS700 can be found here
http://www.apcc.com/resource/include/te ... u=SU700net