"Due to the disruption at French ports, HGVs (heavy goods vehicles) destined for Port of Dover and Eurotunnel terminals are being 'stacked' on the M20," the local Kent police force said in a statement.
One section of motorway was full with 2,300 trucks and around half of a second section with capacity for another 2,300 was also filled as part of a contingency plan named "Operation Stack".
A ticketing system is being implemented and only drivers with tickets can enter Dover itself.
"We sincerely regret the impact to the travelling public, freight and the Dover community of a situation that is beyond our control," a Port of Dover spokesman said.
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"We will continue to monitor the situation closely in liaison with our ferry partners and the Port of Calais in order to resume normal operations as soon as possible," he added.
"The UK and French governments must acknowledge their responsibilities to all Port of Calais users, move in and act. If this means deployment of the armed forces then so be it," he said.
"Let's get this desperate mess sorted out now and talk about a long term solution afterwards. The scale of the current situation has to be seen to be believed," he said.
Dover's local MP Charlie Elphicke said the motorway blockages were isolating his constituents.
"For the record, the people of Dover are now stranded, completely cut off in their own town," the Conservative lawmaker tweeted.