We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on points. We first consider the -space regime and show that, when the input is a stream of all entries of the metric, for any , both MST and TSP cost can be -approximated using space, and that space is necessary for this task. Moreover, we show that even if the streaming algorithm is allowed passes over a metric stream, it still requires space. We next consider the semi-streaming regime, where computing even the exact MST cost is easy and the main challenge is to estimate TSP cost to within a factor that is strictly better than . We show that, if the input is a stream of all edges of the weighted graph that induces the underlying metric, for any , any one-pass -approximation of TSP cost requires space; on the other hand, there is an space two-pass algorithm that approximates the TSP cost to within a factor of 1.96. Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than , when the algorithm is given access to a matrix that specifies pairwise distances between all points. For MST estimation in this model, it is known that a -approximation is achievable with queries. We design an algorithm that performs distance queries and achieves a strictly better than -approximation when either the metric is known to contain a spanning tree supported on weight- edges or the algorithm is given access to a minimum spanning tree of the graph.