It's hard to talk about closeness threshold, since it depends greatly on other parameters (# nodes, # edges,...). 

One thing you can know for sure is that every closeness centrality is somewhere between 2/[n(n-1)] (which is minimum, attained at path) and 1/(n-1) (which is maximum, attained at clique or star).

Perhaps better question would be about Freeman centralization of closeness (which is somehow normalized version of closeness that you can better compare between various graphs).

In connected graphs there is a natural distance metric between all pairs of nodes, defined by the length of their shortest paths. The fairness of a node x is defined as the sum of its distances from all other nodes, and its closeness as the reciprocal of the fairness.

Thus, the more central a node is the lower its total distance from all other nodes. Note that taking distances from or to all other nodes is irrelevant in undirected graphs, whereas in directed graphs distances to a node are considered a more meaningful measure of centrality, as in general (e.g., in, the web) a node has little control over its incoming links.

When a graph is not strongly connected, a widespread idea is that of using the sum of reciprocal of distances, instead of the reciprocal of the sum of distances.

Generally you will want to construct null model(s) that will give you some insight into the significance of the closeness centrality values you extract.  You will compare the distribution of centrality values in your graph with the distribution of the null model(s), which will give you some insight into what these values mean.  As many graph parameters are influenced by the number of nodes and edges, the null model should at the very least account for this e.g. by generating a random graph with an equivalent number of nodes and edges.  However, you may want to control other aspects of network structure that are of no interest, especially if you graph is directed and/or weighted.  This really depends on the specific question that you are trying to answer.  For example, you may want to say that there is something special about the way nodes are connected that is leading to a specific pattern of centrality levels (e.g. a rich club).  In a case like this, you would want to create a null model through permutation/shuffling the node assignments of the graph, which maintains the degree distribution of the graph.  Alternate approaches that could come up are shuffling the weights or directedness of edges. Regardless, carefully think about your specific question--significant compared to what--before moving forward.  Good luck!