For the moment, you can query a node that is in sync. Eventually a block explorer will be available.
Testnet
On testnet, you can query http://146.71.79.179.
First, check if the node is in syncing
curl -X POST --data '{"jsonrpc":"2.0","method":"eth_syncing","params":[],"id":1}' -H "Content-Type: application/json" http://146.71.79.179:8545
If the node is in sync, it will return
{"jsonrpc":"2.0","id":1,"result":false}
Then query for block height
curl -X POST --data '{"jsonrpc":"2.0","method":"eth_blockNumber","params":[],"id":1}' -H "Content-Type: application/json" http://146.71.79.179:8545
The response will include the block height in hex. Convert the hex value to a decimal to determine the block height.