Safe is referring to the monomer and other components also being relatively harmless and not explosive, or extremely reactive to air or other lab environment conditions.
Links to papers and studies showing materials and methods would be appreciated.
First opportunity is to react di- or polycarbonates with di- or polyamines to get poly(hydroxy urethanes) which resemble in properties polyurethanes, the catalysis is to be established based on literature knowledge (there is abundant literature and patents available)
Second opportunity is to react di- or polyepoxydes with di- or polythiols, catalysis see above. The functionality of the raw materials are to be chosen to meet the properties aimed at, elastomers up to rigid foams.
A non toxic way to produce PUs by hand mixing without machines is use of prepolymers. In prepolymers the diisocyanate is already reacted with a polyol and in some prepolymers the excess of non-reacted diisocyanate is distilled off. Such a TDI terminated prepolymer - nearly free of unreacted TDI - is Desmodur E15 from Covestro, with an isocyanate content of 4.4, which means an equivalent weight of 950. A suitable amine hardener of such prepolymers is Ethacure 300 from Albemarle (this is the only amine hardener with a sufficient long pot life for hand mixing) with an equivalent weight of 107. The mixture should have a slight isocyanate exccess - that means a mixture of 1 kg Desmodur E15 and 100 g Ethacure 300 will be the basis of a very elastic cast resin with urethane and urea groups. A cheap alternative for Ethacure 300 which is also available from SIGMA-ALDRICH is triisopropanolamine with an equivalent weight of 64. In this case the cast resin contains only urethane groups but some urea groups are good for mechanical properties.
Referring to Mr. Köhler, there are some other oligomeric isocyanates which can be used in the same way being characterized by an monomer isocyanate content
The first link refers to amino functional PU hardeners and the second and third link to isocyanate terminated prepolymers. You must only combine the isocyanate component and the hardener component in a stoichiometry near 1:1 with a slight isocyanate excess. The easiest way of calculation of stoichiometry is the use of the equivalent weight. 1-3% isocyanate excess is appropriate.