Fast neutron spectrometry and dosimetry at a specified location in a compact radiation environment is difficult due to large size of the conventional neutron spectrometers. In such situations, a small-size bare CR-39 detector can be used as a neutron detector to quantify the fast neutron fraction. In the present study, CR-39 detectors were placed very close to natural lithium and carbon targets during the proton irradiations to acquire the emission neutron spectra. The neutron spectra and the ambient dose equivalents in the forward and lateral directions with respect to the incident proton beam were estimated for both targets at incident proton energies between 8 and 20 MeV. The measured neutron spectra using CR-39 detectors could identify the quasi mono-energetic neutron features from the Li(p,n) and 13C(p,n) reactions effectively. An important observation of the present study is the identification of the fast neutron signature from the 13C(p,n) system from the discrete state de-excitations of excited 14N composite nuclei. The theoretical evaluation of the neutron spectral features and relative neutron energy distributions were performed using the FLUKA: FLUktuierende KAskade, a Monte Carlo simulation package, and the estimates agreed with the experimental results for both systems. The neutron ambient dose equivalents were also estimated from the measured spectra. These neutron fluence and dose estimates at close vicinity to the target can serve as an essential basis for shielding calculations and planning the pertinent radiation protection strategies.