I'm very impressed by the fan-made episodes of Star Trek Continues. I was wondering, since they are so well made, are they considered to be canonical?
What would it take for the episodes to be considered canonical?
Unfortunately, the quality of a piece is not enough to qualify it automatically into a canon.
Usually, the only way something is recognized as being incorporated into canon is if it is officially licenced/produced, but there is a lot of tension between the canonicity of material within fan cultures and 'official' work. That being said, some fan content can be adopted into canon (such as Mark Gruenwald's patient Universe designation cataloguing of Marvel), but this is a rare and interminable process.
Whilst I haven't personally seen the episodes, it would be safe to assume that as they have been produced outside the official franchise this makes them de-facto non-canon (even if they observe the details of and avoid contradicting the canon themselves).
In order for them to be promoted to canonical material, it would take either the deployment of tactical overlap from an official position (meaning an official Star Trek property to reference something directly from it) or an overt statement from the property owners that they have adopted the work into their official universe.
The latter is unlikely, however, as there are no doubt issues of remuneration that would be raised by such a statement. From a ownership position, something doesn't have to be adopted officially in order for the property to benefit from its existence: the fact that the work exists helps to circulate the cultural currency of Star Trek, and so it would make little financial difference if the work is credited by the studios or not.
Acknowledgement of fan work is common, and helps promote the line-blurring of producers/consumers which most fandom sustains, but there is little chance of it being officially adopted into canon.