The newest series of Doctor Who premiered on Saturday, with liberal application of the Doctor's sonic screwdriver. One of the things I find fascinating is how different fans react to how the screwdriver has been used by the different incarnations of the Doctor. Matt Smith (the current Doctor) uses it largely as a non-destructive evaluation tool, or as, well, a screwdriver. The Tenth Doctor, David Tennant, was a bit more reliant on his screwdriver.
In "The Empty Child" Chris Eccleston (the Ninth Doctor) uses it to repair a chain link fence, which one anonymous internet commentator was offended by. But here's the thing: ultrasonic welding actually exists. It can be used on a variety of plastics, as well as metals, and unlike traditional welding, is very effective in joining dissimilar metals. It is often used for applications like microelectronics, where other forms of welding are simply too imprecise.
Fundamentally, ultrasonic welding works by creating a high-pressure wave creating local motion of the material. The mixing at the interface joins the two pieces. While current ultrasonic welding is limited to small thickness in metal, one can assume that given the massive power of the TARDIS, the Doctor can probably manage to weld some chain link fence back together. It's also one of the better processes to effectively join composite materials, without introducing defects, as typically happens with adhesives and definitely happens with mechanical fasteners.
So even though the concept of a time-traveling police box may be far-fetched, there is some science in science fiction.