The problem can be tackled from several points of view. First of all, one may simply use a (unital) $^*$-algebra (the so-called Borchers-Uhlmann algebra in the QFT case), thus dropping any requirement on boundedness of observables, and all major features of the algebraic approach are preserved, like the GNS construction. Though, obviously, several technicalities become more complicated since, the relevant topological properties have to be introduced from other ways (in terms of seminorms possibly induced by a physically sensible class of states).
However, sticking to proper $C^*$-algebras and thus dealing with (abstract) bounded observables, the boundedness requirement is not so physically untenable as it could seem at first glance. Assume, indeed, to work in a given Hilbert space with concrete algebras of operators, and focus on an unbounded observable $A$. Experiments can appreciate only an arbitrarily large but finite range of values $[-n,n]$ of $A$. So, concerning values attained by $A$, it is not possible to distinguish between
$$A = \int_{\sigma(A)} \lambda dP^{(A)}(\lambda)$$
(where we have exploited the spectral decomposition of $A$) and the bounded observable, say:
$$A_n := \int_{[-n,n]\cap \sigma(A)} \lambda dP^{(A)}(\lambda)\:,$$
satisfying $||A_n||\leq n$.
It is possible to distinguish between these two observables relying upon theoretical issues. For instance $A$ (but not $A_n$) may be the generator of a physically relevant unitary symmetry of the considered physical system.
In any cases, the whole class of bounded observables $\{A_n\}_{n\in \mathbb N}$ includes the whole physical and mathematical information of $A$ itself. In particular, mathematically speaking, $A_n \to A$ in the strong operator topology for $n\to +\infty$.
Finally, even starting form an abstract $C^*$-algebra, physically meaningful unbounded observables always arise as soon as one fixes an algebraic state and represent the algebra in the associated GNS Hilbert space. Therein, for instance, all continuous symmetries enjoyed by the state (and represented by $C^*$-algebras automorphisms leaving invariant the state) are (strongly continuously) unitarily implemented and therefore admit (generally unbounded) self-adjoint generators with physical meaning.
All conserved quantities (energy, momentum, etc...), typically represented by unbounded self-adjoint operators, enter the theory this way, concerning for instance the local Weyl $C^*$ algebra of field operators in a QFT, as soon as a reference state is chosen.
(It is worth stressing that the same procedure may give rise to superselection rules, in addition to those already present in the abstract $C^*$-algebras of observables. These are associated with the choice of the reference state where one represent the theory and the von Neumann algebra generated by the GNS representation.)
