This is an intricate and fundamental issue especially to make contact with what really happens in laboratories. Actually, all conceivable measurements are in fact of observables with point spectrum in view of the finite sensibility of the instrument.
So it is correct to speak of eigenvalues in proper sense.
For instance, a measurement of the position along the $x$ axis is performed with an instrument with sensibility $\delta$. This means that we have a sequence (ideally infinite) of detectors able to detect the particle in the segment $x-\delta/2$, $x+\delta/2$. Mathematically speaking, we have a projection valued measure labeled by intervals
$$P([x_i,x_i+\delta))$$
where $x_{i+1}=x_i+\delta$. The union of these intervals is the whole real line.
$P([x_i,x_i+\delta))$ is the elementary YES-NO observable "the particle is found in $[x_i,x_i+\delta)$": an orthogonal projector. Orthogonal projectors associated to disjoint intervals are mutually orthogonal: their product is $0$. The sum (strong topology) of all these projectors is $I$ and an analogous statement is valid regarding the projector associated to a countable union of intervals. The spectrum of each ortogonal projector is $\{0,1\}$. When we measure all them simultaneously, at a given time $t_0$, only one of these tests gives the outcome $1$ ("YES").
The approximate position operator can be defined as
$$\tilde{X} = \sum_n x_nP([x_n,x_n+\delta))$$
(in the strong operatorial topology). $x_n$ may be replaced for any other point in the sensibility interval.
The spectrum of $\tilde{X}$ is a pure point spectrum with elements, all eigenvalues, $x_n$.
In the Hilbert space $L^2(\mathbb{R})$ the projectors above mentioned take the usual form
$$(P([a,b))\psi)(x)= \chi_{[a,b)}(x)\psi(x)$$
where $\chi_{[a,b)}(x)=1$ if $x\in[a,b)$ and $\chi_{[a,b)}(x)=0$ otherwise.
It is possible to prove that, in a suitable topology, this operator tends to the theoretical $X$ when $\delta \to 0$. It is clear that the Heisenberg principle has to be reformulated referring to this approximate setup. Its validity, in the theoretical popular form everybody knows, is valid in the limit of small $\delta$.
A difficult problem is the post measurement state $\psi'$ when we known that the outcome of the measurement is $x_n$ and the pre measurement (pure) state was $\psi$.
Here there are different answers. The common answer which, in a certain technical sense, assumes that all possibilities are equiprobable, is the Lüders projection postulate:
$$\psi' = P([x_n,x_n+\delta))\psi$$
up to normalization.
If instead one knows something more about the effective procedure performed by the instrument, the answer is given in terms of the specific Kraus operator $K_n$ that decomposes the projector.
$$P([x_n,x_n+\delta)) = K^*_nK_n\:.$$
In this case,
$$\psi' = K_n\psi$$
again up to normalization. The results trivially extend to the case of mixed states.
