We consider the spectrum of a U(1) quantum link model where gauge fields are realized as S=1/2 spins and demonstrate a new mechanism for generating quantum many-body scars (high-energy eigenstates that violate the eigenstate thermalization hypothesis) in a constrained Hilbert space. Many-body dynamics with local constraints has attracted much attention due to the recent discovery of nonergodic behavior in quantum simulators based on Rydberg atoms. Lattice gauge theories provide natural examples of constrained systems since physical states must be gauge invariant. In our case, the Hamiltonian H=O_{kin}+λO_{pot}, where O_{pot} (O_{kin}) is diagonal (off-diagonal) in the electric flux basis, contains exact midspectrum zero modes at λ=0 whose number grows exponentially with system size. This massive degeneracy is lifted at any nonzero λ but some special linear combinations that simultaneously diagonalize O_{kin} and O_{pot} survive as quantum many-body scars, suggesting an "order-by-disorder" mechanism in the Hilbert space. We give evidence for such scars and show their dynamical consequences on two-leg ladders with up to 56 spins, which may be tested using available proposals of quantum simulators. Results on wider ladders point towards their presence in two dimensions as well.