Arduino 433Mhz weather sensor receiver
You could use an Arduino with an ethernet shield to store the data to a MySQL server on the local network. See this blog post for an introduction to a library.

The tellstick is designed to be plugged into a USB host, so it doesn't make much sense in combination with an Arduino. You can get a USB host module but it's an expensive solution and you are using way more hardware than necessary.

Instead you could get 433 MHz receivers dirt cheap on ebay, but I'm afraid they only work on one channel, and there are a bunch of channels in the 433 MHz spectrum. So they might not work with the sensors you are talking about.

Another option would be to use eg. Arduino Pro Mini's in combination with a 433 MHz transmitter, powered by a battery. You could get the temperature reading from the internal sensor, or use an external one. The internal sensor takes quite a bit of calibration work to give a precise reading though.
You can get Pro Mini clones on ebay for a few dollars. You would have to dive into sleep modes and such, if you don't want't the batteries eaten away in a short time. To program it, you'll need an FTDI cable, or another Arduino programmed as an ArduinoISP (In System Programmer).
As an alternative to the Pro Mini board you could use a plain AVR chip, but it will be more work.