TYPO3 Core

Two steps to solve this:
1) TCA

'config' => array(
    'type' => 'input',    
    'size' => '8',    
    'eval' => 'time',
    'checkbox' => '',
)

2) JavaScript t3lib/jsfunc.evalfield.js ~line 458

    case "time":
    case "timesec":
//        if (!parseInt(value))    {return '';} // WRONG
        if (!parseInt(value) && value !== '0')    {return '';} // RIGHT
        var theTime = new Date(parseInt(value) * 1000);

There should be an explanation for the checkbox parameter in the documentation, preferably at TCA: type "input", eval "time".

Requesting to include the bugfix in the next version of the core.

In order to make 0:00 work with eval => "time,required", you have to modify typo3/jsfunc.tbe_editor.js too.
~line 209

  if (!value || elementData.additional && elementData.additional.isPositiveNumber && (isNaN(value) || Number(value) < 0)) {

Change "less that or equal zero" to "less than zero".

After that change, the text "isPositiveNumber" does not reflect its behaviour anymore. So that keyword should be renamed as well.

And I am afraid after the changes it is not possible anymore to have null values - maybe there should be done something about that, too.

Patch set 1 for branch master of project Packages/TYPO3.CMS has been pushed to the review server.
It is available at https://review.typo3.org/45058

Applied in changeset 4ddb793fa3f20ac93829783702046631a50bfefc.