Pioneer 10 and Pioneer 11 -- launched in 1972 and 1973 to explore Jupiter and Saturn -- have traveled to the most distant fringes of the solar system. As their fuel was spent decades ago, their travels are subject only to momentum and gravity. Therefore, their paths should be eminently predictable. Problem: They're hundreds of thousands of miles away from where they should be, and deviating by ~8,000 more miles every year.
The Pioneer Anomaly, as it's called, is one of the biggest physics mysteries around. JPL scientists started working in 1980 to figure out why Pioneer 10 and 11 get more and more off-course every year, proposing, testing and eventually discarding various explanations. It may turn out to be that our understanding of physics is due for a serious re-evaluation. Problem #2: NASA is about to destroy our only means of figuring this out.
The main body of data from the two spacecraft exists only on magnetic tapes that can only be read by an obsolete computer system. Access to the tapes (from 1972-1987) would allow researchers to go over every bit of information recorded by the probes. But NASA has pulled funding from further research into the anomaly, and is set to demolish the only computers able to read the data.
The Planetary Society, a group of space exploration enthusiasts, is trying to pull together sufficient money to save the computers from destruction, pull the data, and begin the analysis. The amount needed is startlingly small -- only about $250,000 -- considering that it's looking more and more like the result could be rewriting the laws of physics.
If need be, someone could build a device to read those magnetic tapes. I'm pretty sure I could manage it given the proper tools.