Let $\cal D$ be the set of all $D=1,2,3,\dots$ such that $D\equiv 0,1 \mod(4)$ and $D$ is not a square. For $D\in\cal D$ let $\varepsilon(D)$ denote the smallest number $\varepsilon=\frac{a+b\sqrt D}2$ which is $>1$ and satisfies $a,b\in\mathbb Z$, $a^2-b^2D=4$ (Pell's equation).
Let $\cal E$ denote the set of all such numbers $\varepsilon(D)$ with $D\in\cal D$. My question is about the differences $\varepsilon-\varepsilon'$.
For $T>0$ let $M(T)$ denote the set of all pairs $(\varepsilon,\varepsilon')$ in ${\cal E}\times\cal E$ such that 
$0<\varepsilon-\varepsilon' < T$.


1. Is it true that there exists $T>0$ such that $M(T)$ is finite?
2. Is it true that $M(T)$ is always finite?


All hints to the literature on properties of the fundamental units $\varepsilon\in\cal E$ are welcome! 



Differences between fundamental units
*Note:* The following answer assumes that the question means (all) fundamental units in its standard meaning as opposed to a litteral reading of the question in the body (see Impossible Hippopotamus's answer for details related to this, and also for a more complete answer in general).


For the second question, it is not true that $M(T)$ is always finite. 


A way to answer this second question is via a result of Sprindžuk, The distribution of the fundamental units of real quadratic fields, Acta Arithmetica (1973/74). There it is proved, among other things, that the number of $\epsilon$, fundamental units of real quadratic number field, with $\epsilon \le x$ is $2x + O(x^{1/2})$, in fact something more precisely is established. 


Thus for every $T\_0 > 1/2$ there need to be infinitely many pairs at most $T\_0$ apart, else one could not have about $2x$ of them in the interval $[1,x]$ (for $x$ ever larger). 


So for $T\_0> 1/2$ one has $M(T\_0)$ is infinite. 


For the first, I do not know. There are some ways to construct explictly sequences of fundamental units, but I did not come across one where the gap is arbitrarily small (but my knowledge here is certainly not at all exhaustive); I suspect however they could be used to give a different argument for the second question. See for example Degert, Über die Bestimmung der Grundeinheit gewisser reell-quadratischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg (1958) 


Somehow I would guess that $M(T)$ is in fact always infinite, based on the fact above that they are not too sparse globally, so somehow sometimes they should be arbitrarily close together. But this is a vague guess and might as well be wrong. 


