<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2018-07-05</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000917470</issuerCik>
        <issuerName>OLYMPIC STEEL INC</issuerName>
        <issuerTradingSymbol>ZEUS</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001255662</rptOwnerCik>
            <rptOwnerName>MARABITO RICHARD T</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O OLYMPIC STEEL, INC.</rptOwnerStreet1>
            <rptOwnerStreet2>22901 MILLCREEK BLVD., SUITE 650</rptOwnerStreet2>
            <rptOwnerCity>HIGHLAND HILLS</rptOwnerCity>
            <rptOwnerState>OH</rptOwnerState>
            <rptOwnerZipCode>44122</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Chief Financial Officer</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Restricted Stock Units</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <footnoteId id="F1"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2018-07-05</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>822</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F2"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F2"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>822</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>54463</value>
                    <footnoteId id="F3"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Each restricted share unit (&quot;RSU&quot;) represents the contingent right to receive one share of Olympic Steel, Inc. common stock.</footnote>
        <footnote id="F2">These RSUs were acquired under Olympic Steel's Amended and Restated 2007 Omnibus Incentive Plan. These RSUs vest on July 5, 2023 if the Reporting Person continues to be an employee of Olympic Steel (or its subsidiaries or affiliates) in the same or a higher position on such date. Any vested RSUs will be settled within 30 days following the first to occur of the Reporting Person's death, disability or retirement after the Reporting Person has attained the age of 62. Vesting and settlement may also occur earlier in limited circumstances as specified in the award agreement.</footnote>
        <footnote id="F3">This total number of RSUs includes previously-granted RSUs with different vesting and payment terms.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Gretchen A. Sterling, as Attorney-in-Fact</signatureName>
        <signatureDate>2018-07-09</signatureDate>
    </ownerSignature>
</ownershipDocument>
