$\def\Z{\mathbf{Z}}$
$\def\eps{\epsilon}$

Observe that if a unit $\epsilon \in L$ lives in a quadratic subfield, then $N(\epsilon) = + 1$. So the index of the units is positive whenever $L$ has a unit of norm $-1$. If $L = \mathbf{Q}(\sqrt{5},\sqrt{13})$, then

$$\epsilon = \frac{9 + \sqrt{5} + \sqrt{13} + \sqrt{65}}{4}$$

is a root of the polynomial $x^4 - 9 x^3 + 20 x^2 - 7 x - 1$, from which one can see that $\epsilon$ is a unit of norm $-1$.

More generally, the index will always be a (small, bounded) power of two (proof, the units of $L$ inject into the units of the subfields under the composition of the three norm maps, whereas the subgroup of units coming from subfields under this map has index $8$). This problem was studied by Kubota. He proved a relationship between the index of this unit group and the ratio of the class number of $L$ to the product of the class numbers of the quadratic subfields. (explicitly, the index is $4h/h_1h_2h_3$.)

There is a nice discussion of this in Section 7 of Maria Stadnik's paper here:

http://arxiv.org/pdf/1202.3475v1.pdf,

from which one learns the following nice generalization of the computation above (see Proposition 7.6): if $p$ and $q$ are primes such that $p \equiv q \equiv 1 \mod 4$ and the Legendre symbol $\left(\frac{p}{q}\right) = -1$, then $\mathbf{Q}(\sqrt{p},\sqrt{q})$ *always* has a unit of norm $-1$, and so the index is always $>1$ (in fact, it will always be $2$ in this case.)

Let me also quote the following from the paper above concerning the work of Kubota: if the fundamental units of the subfields of $L$ are $\epsilon_1$, $\epsilon_2$, and $\epsilon_3$, then Kubota "completely classified the possible structures into one of seven types." As an example, "if every subfield has a unit of norm $-1$, then the units of $L$ are either of the two forms $\{\epsilon_1,\epsilon_2,\epsilon_3\}$ or $\{\epsilon_1,\epsilon_2,\sqrt{\epsilon_1, \epsilon_2, \epsilon_3}\}$", the latter occurring if and only if $L$ has a unit of norm $-1$.

In the imaginary case, if $\epsilon$ is the fundamental unit of $L$ (the unit rank is one) and $K$ is the quadratic subfield, then $N_{L/K}(\epsilon)$ is a unit of $K$. On the other hand, if $\eta$ is the fundamental unit of $K$, then, thought of as an element of $L$, $N_{L/K}(\eta) = \eta^2$. Hence, up to roots of unity in $L$, either $\eta$ is the fundamental unit of $L$ or is a square of the fundamental unit. If the latter occurs, then $\eta \zeta$ is a square in $L$ for a root of unity $\zeta \in L$. Both cases may happen:

In $L = \mathbf{Q}(\sqrt{-1},\sqrt{5})$, the unit group is $\mu_4 \oplus \Z$, where $\Z$ is generated by the fundamental unit of $\mathbf{Q}(\sqrt{5})$.

In $L = \mathbf{Q}(\sqrt{-1},\sqrt{6})$, the unit group is $\mu_4 \oplus \Z$, where $\Z$ is generated by $\displaystyle{\eps = \displaystyle{\frac{1-\sqrt{-1}}{2}} \cdot (2 + \sqrt{6})},$

and $\eps^2 = \zeta \eta$, where $\zeta^4 = 1$ and $\eta = (5 + 2 \sqrt{6})$ is the fundamental unit of $\mathbf{Q}(\sqrt{6})$. (So the relevant indices in these two cases are two and four respectively.)