The appearance of the scroll bars can be controlled with WebKit’s -webkit-scrollbar pseudo-elements ^[blog]. You can disable the default appearance and behaviour by setting -webkit-appearance ^[docs] to none.

Because you’re removing the default style, you’ll also need to specify the style yourself or the scroll bar will never show up. The following CSS recreates the appearance of the hiding scroll bars:

Example (jsfiddle)

CSS

.frame::-webkit-scrollbar {
    -webkit-appearance: none;
}

.frame::-webkit-scrollbar:vertical {
    width: 11px;
}

.frame::-webkit-scrollbar:horizontal {
    height: 11px;
}

.frame::-webkit-scrollbar-thumb {
    border-radius: 8px;
    border: 2px solid white; /* should match background, can't be transparent */
    background-color: rgba(0, 0, 0, .5);
}

.frame::-webkit-scrollbar-track { 
    background-color: #fff; 
    border-radius: 8px; 
} 

WebKit (Chrome) Screenshot