Could computer architecture be reworked to support switching from binary to ternary, for example? I was investigating why computers use binary instead of ternary and, from what I understand, found it's for precision. We want to avoid getting the wrong answer when computing math, and if we were to look at a bit at the wrong time in, for example, a base-10 system we might accidentally catch it switching states and accept the output before it's done switching states.

Please let me know if I've misunderstood this. I know there's a lot going on under the hood in computer hardware.

Assuming I understood correctly though, would it be feasible to redesign computers to be able to switch "languages" in some components in real-time in favour of speed/size instead of accuracy? It's well-known that 8 bits in base-10 can store 390,625 times more data than in binary, so there are obviously massive implications if possible.

To clarify: my question is about the feasibility of designing a computer which could switch between the two languages in real-time based on user-defined required output.

Edit: As clarified above, I am not asking about the benefits of binary over ternary. I am asking about the benefits of using them together, in real-time, and if that could even be possible. Thanks.
Current computers use electronic parts which can easily distinguish between two states, by stating that high voltage = 0 and low voltage = 1, or vice versa. The hardware that is needed to combine two inputs x and y and produce an output equal to NAND (x, y) is very, very simple and any other logic can be produced very easily from this. 

Your task in proposing ternary computers: Suggest how with current electronic parts, three states can be represented. Then suggest what would be a primitive operation that combines two ternary inputs and produces an output; what that operation would be, and how you would implement it. Then suggest how you could for example add two integer numbers that are made of 40 trits (I think that's the name, I might be wrong). 

Of course your suggestion must not represent a trit using two binary states, because using two binary states you could easily represent four values with the same effort, which would surely be more efficient. So you might represent 0 = low voltage, 1 = medium voltage, 2 = high voltage. Have fun doing that reliably. (One problem that you have is that using low and high only, you can change states with force and therefore quickly. If you needed to change a state from high to medium voltage, you'd have to be careful not to overshoot to low voltage, so this would likely be slower). 

Now it seems you not only want to implement ternary arithmetic, but you want to switch the hardware on the fly between binary and ternary. That's not running before you can walk, that's running before you even figured out that crawling might be a possibility.