Knowledge graphs and ontologies are becoming increasingly important as technical solutions for Findable, Accessible, Interoperable, and Reusable data and metadata (FAIR Guiding Principles). We discuss four challenges that impede the use of FAIR knowledge graphs and propose semantic units as their potential solution. Semantic units structure a knowledge graph into identifiable and semantically meaningful subgraphs. Each unit is represented by its own resource, instantiates a corresponding semantic unit class, and can be implemented as a FAIR Digital Object and a nanopublication in RDF/OWL and property graphs. We distinguish statement and compound units as basic categories of semantic units. Statement units represent smallest, independent propositions that are semantically meaningful for a human reader. They consist of one or more triples and mathematically partition a knowledge graph. We distinguish assertional, contingent (prototypical), and universal statement units as basic types of statement units and propose representational schemes and formal semantics for them (including for absence statements, negations, and cardinality restrictions) that do not involve blank nodes and that translate back to OWL. Compound units, on the other hand, represent semantically meaningful collections of semantic units and we distinguish various types of compound units, representing different levels of representational granularity, different types of granularity trees, and different frames of reference. Semantic units support making statements about statements, can be used for graph-alignment, subgraph-matching, knowledge graph profiling, and for managing access restrictions to sensitive data. Organizing the graph into semantic units supports the separation of ontological, diagnostic (i.e., referential), and discursive information, and it also supports the differentiation of multiple frames of reference.