Back when I was a teaching assistant, we were teaching utility theory and how it relates to price determination. So we were looking at continuous goods (e.g. "food") so as to get marginal utility. But a student asked about buying a bed; the choice is zero or one, a person doesn't need two or more. So is it possible to determine marginal utility in this case, and if so how?
Yes, it is. First, what you describe is not as much binary choice, but situation where you have discrete quantities where any quantity higher than 1 does not bring any benefits (a person can have 2 beds just the second bed is useless). You can calculate marginal utility for such case normally how you would do for other goods that come in discrete quantities (I think you might be abusing the word continuous here since many intro micro courses, assume food is discrete not continuous).
For a discrete quantity, marginal utility, is the utility of the additional of the last unit consumed, hence in discrete case
$$MU = U(n+1) - U(n)$$
Where $n$ is the number of units already consumed.
For example, using slices of pizzas, if 0 slices of pizza give you 0 utility, 1 slice of pizza gives you utility of 10 and 2 slices of pizza gives you utility of 15 then we have:
Slices of Pizza U MU 0 0 NA 1 10 10 2 15 5
Now if we assume that people only get utility from just having one bed and would never want to buy more, so we assume for $U(bed)$ that $U(0)=0$, $U(1)=10$, $U(2)=10$ and $U(3)=10$ ... then we have
Beds U MU 0 0 NA 1 10 10 2 10 0 3 10 0 ...
As you can see above the additional beds just do not add any additional utility so MU is always zero (except from moving from zero beds to one bed), but you can certainly calculate it for discrete case.