Thanks again for your additional comments. After thinking about my goal overnight (while I was asleep !), I think I have been making this project too complicated. What I really need is to be able to check the temperature in the remote building at will. To do that, all I need is a serial modem attached to an Arduino that has a temperature sensor.
The Arduino would have to monitor and answer my incoming call (through the serial modem) and respond with the current temperature. This would serve my purpose and would avoid any problems either with power or with data storage on the Arduino. If I wanted to store temperature data and/or call repeatedly, I could do that with the calling computer.
During operation, if I get a response from the Arduino with the appropriate temperature, then the situation is OK. If I get a response with a low temperature, then I need to investigate the situation. If I get no response at all, either due to a power failure or some other cause, then I still need to investigate. That arrangement would serve the purpose that I am trying to address. A few additional comments:
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The remote building is about 3/4 mile from the main office, so I think that's too far for some of the ideas mentioned previously.
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I do want a positive response from the system on demand, not just an error message when the temperature is outside of the limit, since the absence of an error message doesn't indicate whether some other problem (such as power failure) has occurred.
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If I wanted to find a low power Linux computer on ebay, what's the best way to search for it?
So I think my next step is to try to find a cheap, working serial modem. I'll look on ebay as was previously mentioned. If anybody knows of any other source, please let me know. I want to try to avoid the expensive ones. (I've got a Arduino Duemilanove on order but it's not here yet.) Thanks again.