How much string content can I store in an option?

Look at the table schema in wp-admin/includes/schema.php:

// regular blog tables
CREATE TABLE $wpdb->options (
  option_id bigint(20) unsigned NOT NULL auto_increment,
  option_name varchar(64) NOT NULL default '',
  option_value longtext NOT NULL,
  autoload varchar(20) NOT NULL default 'yes',
  PRIMARY KEY  (option_id),
  UNIQUE KEY option_name (option_name)
) $charset_collate;

// Multisite options
CREATE TABLE $wpdb->sitemeta (
  meta_id bigint(20) NOT NULL auto_increment,
  site_id bigint(20) NOT NULL default '0',
  meta_key varchar(255) default NULL,
  meta_value longtext,
  PRIMARY KEY  (meta_id),
  KEY meta_key (meta_key),
  KEY site_id (site_id)
) $charset_collate;

In both cases the value is longtext. The MySQL manual says about that type:

A TEXT column with a maximum length of 4,294,967,295 or 4GB (2³² – 1) characters. The effective maximum length is less if the value contains multi-byte characters. The effective maximum length of LONGTEXT columns also depends on the configured maximum packet size in the client/server protocol and available memory. Each LONGTEXT value is stored using a 4-byte length prefix that indicates the number of bytes in the value.

4GB. If that is not enough improve the clean-up process. Maybe you should store the content as a file anyway – in the uploads directory.