I've been using POE outdoor receiver systems for ADS-B with virtually identical components to Outernet (RPi, RTL-SDR, LNA) for over a year with no issues. It's quite reliable. We have to keep in mind that this hobby gear is in no way anywhere near the sort of build or durability of a commercial outdoor access point or bridge, IP camera, etc.
With Outernet, the signal integrity is somewhat more fragile than ADS-B, and there is the added issue of heat sensitivity. A ventilated ADS-B system runs fine at 50-60C all day long. A loss of sensitivity there means a few less planes spotted, while for Outernet it can mean total lack of decoding, rendering the system useless
For a "permanent" home installation, a suitable NEMA IP66 enclosure for the Antenna/LNA combo can be achieved for relatively low cost, around $15. Then the RTL/CHIP can be kept indoors in a controlled environment. A ventilated project box would suffice, with a good fan or heatsink/fan for the RTL. The total cost for "remoting" the Antenna/LNA is probably around $40-$50 after you include the coaxial cable and connectors, unless you have these items in the junk box.
A complete system in an outdoor enclosure could be accomplished with some creative airflow engineering. Making it POE adds another significant heat source to the mix unless there is a CHIP POE "shield" that can handle the RTL load and run cooler than a typical POE splitter.