In addition to prefixing the X column type with a \hspace{0pt} directive to enable hyphenation -- see also David's answer -- you may want to create a list of acceptable hyphenation patterns, as the babel/ngerman combination tends not be informed all that well about how words such as "Polyvenylchlorid" and "Polyethafluorethen" might be hyphenated acceptably.
\documentclass[a4paper, 12pt]{scrartcl}\usepackage[T1]{fontenc} % to list words with Umlauts in arg. of '\hyphenation'\usepackage[ngerman]{babel}\usepackage{tabularx,ragged2e}\newcolumntype{L}{>{\hspace{0pt}\RaggedRight}X}\renewcommand\tabularxcolumn[1]{m{#1}}\usepackage{chemfig}% Create a list of hyphenation exceptions:\hyphenation{plastik-tüten poly-venyl-chlorid poly-etha-fluor-ethen flaschen-deckel einsatz-beispiel venyl-boden venyl-böden schall-platten}\begin{document}\begin{tabularx}{\textwidth}{|L|c|c|L|} Name & Monomer & Polymermolekül & Einsatzbeispiel\\ \hline Polyethen (PE) & \chemfig{ H -[1]C (-[3]H) =C ( -[1]H) -[7]H } & \schemestart \chemleft[ \chemfig{ -C ( -[2]H) ( -[-2]H) -C ( -[2]H) ( -[-2]H) } \chemright]$_n$ \schemestop& Plastiktüten \\ Polypropen (PP) & \chemfig{ H -C ( -[2]H) ( -[6]H) -C ( -[2]H) =C ( -[1]H) ( -[7]H) } & \schemestart \chemleft[ \chemfig{ -[0]C ( -[2]H) ( -[-2]H) -C ( -[2]{CH3}) -[0] } \chemright]$_n$ \schemestop& Flaschendeckel, Brotdosen \\ Polyvenylchlorid (Pvc)& \chemfig{ H -[1]C ( -[3]H) =C ( -[1]Cl) -[7]H } & \schemestart \chemleft[ \chemfig{ -[0]C ( -[2]H) ( -[-2]H) -C ( -[6]H) ( -[2]{Cl}) -[0] } \chemright]$_n$ \schemestop& Rohrleitungen, Venylböden, Schallplatten \\ Polyethafluorethen (PTFE) & \chemfig{ F -[1]C ( -[3]F) =C ( -[1]F) -[7]F }& \schemestart \chemleft[ \chemfig{ -C ( -[2]F) ( -[6]F) -C ( -[2]F) ( -[6]F) -[0] } \chemright]$_n$ \schemestop& Pfannenbeschichtung (Tephlon), Funktionskleidung (Gortex) \end{tabularx}\end{document}