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FU Orionis (FUor) and EX Lupi (EXor) type objects represent two small, but rather spectacular groups of low-mass, young eruptive stars. Outbursts of several magnitudes are observed, attributed to enhanced accretion from the circumstellar disk onto the central protostar. The host molecular environments of FUors/EXors are poorly explored due to the scarcity of systematic molecular line observations. We carried out the first dedicated survey of the molecular environments of a large sample of FUors/EXors, observing a total of 51 sources with the aim of studying the ammonia (NH$_3$) emission in their host environments. We observed the ammonia (J,K)=(1,1), (2,2), and (3,3) inversion transitions using the Effelsberg 100-m radio telescope. We derived H$_2$ column densities and dust temperatures using archival Herschel SPIRE data. We detected the (1,1) transition toward 28 sources and the (2,2) transition toward 12 sources, while the (3,3) transition was detected toward only two sources. We find kinetic temperatures between ~12 K and 21 K, ammonia column densities from $5.2\times10^{13}\,cm^{-2}$ to $3.2\times10^{15}\,cm^{-2}$, and fractional ammonia abundances with respect to H$_{2}$ from $4.7\times10^{-9}$ to $1.5\times10^{-7}$. The results are comparable to those found in infrared dark clouds (IRDCs). Kinetic analysis suggests that most of the eruptive stars in our sample reside in rather quiescent (sonic or transonic) host environments. Our NH$_3$ observations and analysis of the SPIRE dust-based H$_2$ column density maps confirm the presence of dense material toward 7 sources in our sample; additional sources might also harbour dense gas based on their NH$_2$ (2,2) detections, might indicate an earlier phase than originally classified. Based on our results, we suggest observations targeting additional molecular lines would help to refine the evolutionary classification of eruptive stars., 20 pages, 2 figures, accepted for publication in A&A