【Besiege】 , 【beleaguer】 , 【invest】 , 【blockade】 mean to surround an enemy in a fortified or strong position so as to prevent ingress or egress.

【Besiege】 implies a sitting down before the entrances to a fortified place (as a castle or a walled town), and it may be used to denote the operation of attackers of a strongly fortified or naturally protected position of the enemy and usually implies the surrounding of it and frequent assaults upon it in order to break down the resistance of the enemy.

【Beleaguer】 does not materially differ from 【besiege】  in meaning, although it springs from a different type of warfare, one where fortifications are less the objects of attack than open cities or occupied positions.

【Beleaguer】 suggests the presence of camps and therefore of great numbers of troops; often it carries the connotations of besetting or harassing rather than of confining or imprisoning.

In extended use 【besiege】 stresses a blocking up or an assailing with importunities and  【beleaguer】 emphasizes a pestering or annoying.

【Invest】 comes very close to 【besiege】 , but it does not, by comparison, carry as strong an implication of strength or of persistence in attack.

In many cases it implies the use of men and weapons to prevent ingress or egress, but it carries little suggestion of frequent assaults upon the position.

【Blockade】  usually stresses a closing of all sea-lanes to those who wish to enter or leave hostile territory. The term usually implies the use of ships or mines to attain this end, but if the attacking country is sufficiently strong, it may imply prohibition of neutral or enemy vessels entering or leaving and efforts to seize or detain those that disobey.

The object of blockading is usually to starve the enemy or to prevent the entrance of essential supplies.